Question

\(A\) function \(f\) is even if \(f(-x)=f(x)\) for all \(x\) in the domain of \(f\). Suppose \(f\) is even. with \(\lim _{x \rightarrow 2^{+}} f(x)=5\) and \(\lim _{x \rightarrow 2^{-}} f(x)=8 .\) Evaluate the following limits. a. \(\lim _{x \rightarrow-2^{+}} f(x)\) b. \(\lim _{x \rightarrow-2^{-}} f(x)\)

Short answer

Answer: a. The limit as x approaches -2 from the right is 5. b. The limit as x approaches -2 from the left is 8.

Step-by-step solution

Use the definition of an even function

We know that an even function is defined by f(-x) = f(x). We will use this property to determine the given limits when x approaches to -2 by replacing x with -x.

Substitute values for limit x->(-2+)

Given \(\lim_{x \rightarrow 2^{+}} f(x)=5\), we will substitute -x for x in this limit: \(\lim_{(-x) \rightarrow -2^{+}} f(-x)\) Now replace the -x in the expression with y, Let y=-x: \(\lim_{y \rightarrow 2^{-}} f(y) = 5\) The limit as x approaches -2 from the right is 5.

Substitute values for limit x->(-2-)

Given \(\lim_{x \rightarrow 2^{-}} f(x)=8\), we will substitute -x for x in this limit: \(\lim_{(-x) \rightarrow -2^{-}} f(-x)\) Now replace the -x in the expression with y, Let y=-x: \(\lim_{y \rightarrow 2^{+}} f(y) = 8\) The limit as x approaches -2 from the left is 8. We have found the limits: a. \(\lim_{x \rightarrow -2^{+}} f(x) = 5\) b. \(\lim_{x \rightarrow -2^{-}} f(x) = 8\)

Key concepts

Calculus

Calculus is a branch of mathematics focusing on limits, functions, derivatives, integrals, and infinite series. It's the study of how things change and allows us to understand the behavior of functions and sequences in mathematical terms. In the realm of calculus, limits are a fundamental concept. They help us understand the behavior of functions as they approach a particular point.

For example, as students approach calculus problems, they'll often find themselves calculating limits to understand how a function behaves near a specific input. This specific exercise on even functions and their limits is an excellent illustration of applying calculus concepts to solve real-world mathematical problems.

Limits of Functions

The concept of a limit in calculus is essential for understanding the behavior of functions as they approach a particular value. Limits can tell us the value that a function approaches as the input, or x-value, gets closer to some number. The notation \(\lim_{x \rightarrow c}f(x)\) is used to denote the limit of function f(x) as x approaches the value c.

When dealing with functions, limits can be approached from both the left (denoted as \(x \rightarrow c^{-}\)) and the right (denoted as \(x \rightarrow c^{+}\)). Understanding both of these one-sided limits is crucial for thoroughly grasping the function's behavior near the point c.

Even and Odd Functions

In mathematics, functions can be classified as even, odd, or neither. An even function is symmetric about the y-axis, which means for every x in the function's domain, \(f(-x) = f(x)\). Graphically, if you fold the graph along the y-axis, both sides would match perfectly. Conversely, an odd function is symmetric about the origin, satisfying \(f(-x) = -f(x)\) for all x in its domain.

These properties are not just visually interesting; they have practical applications when solving problems involving limits. For even functions, knowing the limit at a positive value of x gives us direct insights into its negative counterpart and vice versa, because of their inherent symmetry.

One-sided Limits

One-sided limits examine the behavior of functions as they approach a specific point from one side only—either from the left or the right. In notation, \(\lim_{x \rightarrow c^{+}}f(x)\) represents the function approaching c from the right, while \(\lim_{x \rightarrow c^{-}}f(x)\) represents approaching from the left.

These are particularly useful when the function has a different behavior on each side of the point, or when the function is not defined at the point itself. For even functions, we can leverage their symmetry to connect the one-sided limits at \(x\) with those at \( -x\), simplifying the process of finding these limits.

Limit Properties

Limits are governed by several properties that allow us to analyze and calculate them more efficiently. For instance, limits are transitive; if \(\lim_{x \rightarrow c}f(x) = L\) and \(\lim_{x \rightarrow c}g(x) = L\), then \(f(x)\) and \(g(x)\) approach the same limit as \(x\) approaches \(c\).

We can also add, subtract, multiply, and divide limits; if \(\lim_{x \rightarrow c}f(x)\) and \(\lim_{x \rightarrow c}g(x)\) exist, then the limit of the sum, difference, product, or quotient (given the denominator isn't 0) is the sum, difference, product, or quotient of the limits. These properties simplify complex limit calculations, as was leveraged in the textbook problem to quickly determine the required limits.